Comment on 'On computing quantum waves exactly from classical action'
Comment on 'On computing quantum waves exactly from classical action'
This comment demonstrates that the recent paper by Lohmiller & Slotine (Proc. R. Soc. A 482: 20250413) contains a foundational mathematical error. By neglecting spatial derivatives of the probability density amplitude, the authors omit the quantum potential (identified by Madelung and Bohm). Their proposed equivalence is not exact but constitutes the standard semiclassical approximation. The illustrative examples either belong to a class where the quantum potential vanishes identically, or import quantum eigenfunctions through initial conditions.
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4.2 out of 5. Audited on TheoryOfEverything.ai. Permanent record: https://theoryofeverything.ai/review-profile/paper/c3e0869a-4314-4411-9b64-541b7ee954df
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This comment presents a methodologically sound critique of Lohmiller & Slotine's claimed exact quantum-classical equivalence. The specialist panel unanimously recognized the mathematical validity of the central correction: substituting ψ = √ρ exp(iφ/ℏ) into the Schrödinger equation necessarily produces the quantum potential term Q = -(ℏ²/2M)(∇²√ρ)/√ρ unless spatial derivatives of the amplitude vanish. The comment correctly identifies that neglecting these derivatives reduces the purported 'exact' method to the standard semiclassical approximation. The derivation in Section 2 is mathematically rigorous and verifiable, with proper application of product rules and correct dimensional analysis. The categorization of the original paper's examples into 'trivial density' cases (where Q vanishes geometrically) and 'circular reasoning' cases (where quantum eigenfunctions are imported via initial conditions) provides valuable diagnostic insight. However, the math specialist flagged five mathematical risk locations where claims are compressed or incompletely demonstrated, particularly regarding the example-specific analyses in Section 3. The assertion that omitting amplitude derivatives is equivalent to assuming ∇√ρ = 0 everywhere is stated more strongly than necessary—the actual requirement is ∇²√ρ = 0 where defined. The comment succeeds as a targeted mathematical rebuttal while maintaining appropriate scope for the genre.
This review was generated by AI for research and educational purposes. It is not a substitute for formal peer review. All analyses are advisory; publication decisions are based on numerical score thresholds.
The comment is largely internally coherent. Its core thesis is stable throughout: substituting ψ_j = √ρ_j e^{iφ_j/ℏ} into the Schrödinger equation necessarily produces the quantum-potential term Q = -(ℏ²/2M)(∇²√ρ_j)/√ρ_j, so any derivation yielding only the classical Hamilton–Jacobi equation must have omitted spatial derivatives of the amplitude. Section 1 states this claim, Section 2 carries out the derivative expansion, and Section 4 returns to the same conclusion. The logical dependence is clear and consistent.
There are, however, a few overstatements and local ambiguities that keep the score below 5. First, the text says that the target derivation 'implicitly assumes ∇√ρ_j = 0' and equates this with a perfectly homogeneous density across all space. Mathematically, the condition needed to remove Q is weaker: ∇²√ρ_j = 0 (away from nodes/singularities), and there are nonconstant harmonic amplitudes such as √ρ ∝ 1/r in 3D free space. The author partly acknowledges this later in Section 3.1, which softens the issue, but this does create a local mismatch between the strong claim in Section 2 and the broader class of vanishing-Q examples in Section 3. Second, the phrase 'unit metric M' is imprecise, since M is then used algebraically as a scalar mass parameter. These are patchable presentation issues rather than core contradictions.
The main mathematical derivation in Section 2 is valid. Starting from ψ_j = √ρ_j e^{iφ_j/ℏ}, the gradient and Laplacian are expanded correctly by standard product and chain rules: ∇ψ_j = (∇√ρ_j + (i/ℏ)√ρ_j∇φ_j)e^{iφ_j/ℏ}, and ∇²ψ_j = [∇²√ρ_j + (2i/ℏ)∇√ρ_j·∇φ_j + (i/ℏ)√ρ_j∇²φ_j - (1/ℏ²)√ρ_j(∇φ_j)^2]e^{iφ_j/ℏ}. Substituting into the Schrödinger equation and separating real and imaginary parts yields, respectively, the continuity equation and the quantum Hamilton–Jacobi equation with the Bohm/Madelung quantum potential. This is a standard and correct calculation, and it directly supports the paper's central objection.
The score is not 5 because several consequential claims are asserted rather than fully proved. Most importantly, the paper says 'without loss of generality' it restricts to a scalar potential with unit metric/scalar M, but does not derive the corresponding result for the more general setting of the target paper. This does not invalidate the displayed derivation, but it leaves a gap in the claimed scope of the rebuttal. In addition, Section 3's analysis of the target paper's examples is compressed: the double-slit density model is schematic, and the claims that the harmonic-oscillator and hydrogen examples import quantum eigenfunctions through initial conditions are plausible but not mathematically established in detail from the cited constructions. These gaps affect the breadth and some application-specific criticisms, but they do not overturn the central mathematical point that omitting spatial amplitude derivatives removes the quantum potential and thus cannot justify an exact general equivalence.
For a comment paper, the central claim is quite testable: one can directly substitute the disputed ansatz into the Schrödinger equation and check whether the Laplacian acting on √ρ generates the quantum-potential term. This provides a clear criterion for refuting either the target paper's exact-equivalence claim or the present comment's objection. The paper also gives discriminating conditions: if ∇²√ρ/√ρ is nonzero for a proposed exact example, then the original classical-action-only derivation cannot be exact as stated; if examples succeed only when Q vanishes or when quantum eigenfunction content is encoded in the initial data, that supports the rebuttal. The limitation is that the submission is not itself proposing a new physical framework with multiple novel quantitative observational predictions; its testability is mainly mathematical/computational and comparative rather than experimental. Still, as a critique of an exactness claim, it is clearly falsifiable in principle and in practice.
The submission is very clear. It states the disputed claim, identifies the alleged mathematical mistake, performs the relevant substitution in a compact but readable way, and explains the consequence in standard language familiar to graduate-level readers. The organization is strong: introduction, mathematical correction, then diagnosis of why the original paper's examples may appear successful. Notation is consistent, and the distinction between total derivatives along trajectories and spatial derivatives in the Schrödinger PDE is communicated sharply. The only minor caveat is that the example-specific critiques are more programmatic than fully worked through in exhaustive technical detail, but that does not materially obstruct comprehension of the main argument.
The core mathematical point—that Madelung substitution yields a quantum-potential term unless amplitude derivatives vanish—is standard and not novel by itself. Likewise, identifying omission of that term as equivalent to a semiclassical/WKB-type truncation is established physics. The paper's contribution lies in applying this standard diagnostic specifically to the Lohmiller-Slotine construction and organizing their examples into two failure modes: special geometries with vanishing Q, and apparent recovery of quantum results via quantum basis information hidden in initial conditions. That targeted critical synthesis has value and is specific, but it is not a new mechanism or new framework. Thus the work is moderately original as a focused rebuttal, not highly original at the level of theoretical invention.
The argument is fully developed with all necessary mathematical steps shown. The paper: (1) clearly states the error in the original work (neglecting spatial derivatives), (2) provides the complete correct derivation including the quantum potential term, (3) systematically explains why each example in the original paper appears to work despite the error, (4) addresses both trivial density cases and the circular reasoning in bound state examples, and (5) places the critique in proper historical context with appropriate references to Madelung and Bohm. All variables are defined, boundary conditions are addressed through the specific examples, and the limitations of the original approach are explicitly stated.
Key Equations (3)
Time-dependent Schrödinger equation (scalar potential V, unit metric M) to be satisfied by the ansatz.
Quantum Hamilton–Jacobi equation (real part) for the phase φ_j; includes the quantum potential term that is missing if amplitude spatial derivatives are discarded.
Definition of the quantum potential Q (Madelung/Bohm term) responsible for quantum corrections to classical energy conservation.
Other Equations (4)
Continuity equation (imaginary part) satisfied by the classical density and action.
Full Laplacian acting on the ansatz, exhibiting the term that yields the quantum potential when separated into real and imaginary parts.
Spatial gradient of the ansatz; shows contributions from amplitude gradients and phase gradient.
Ansatz constructing the quantum wave function from a classical density amplitude and classical action/phase along path j.
Testable Predictions (2)
The Lohmiller & Slotine construction ψ_j = √ρ_j e^{iφ_j/ℏ} is not an exact solution of the Schrödinger equation for general localized quantum states with nonzero spatial amplitude gradients; it only becomes exact when the quantum potential Q vanishes.
Falsifiable if: Exhibit a nontrivial quantum state with spatially varying amplitude (so Q ≠ 0) for which substituting ψ_j = √ρ_j e^{iφ_j/ℏ} into the Schrödinger equation yields exact equality (both real and imaginary parts satisfied) without invoking additional quantum input.
The examples in Lohmiller & Slotine that reproduce quantum results do so either because the quantum potential vanishes (e.g., plane waves, 1/r densities) or because the initial density decomposition implicitly imports quantum eigenfunctions; they therefore do not validate an exact classical-to-quantum equivalence.
Falsifiable if: Provide an example treated by the same method where the quantum potential is nonzero, and show the method reproduces full quantum dynamics and spectra without choosing initial conditions or expansions that assume the quantum eigenbasis.
Tags & Keywords
Keywords: quantum potential, Madelung hydrodynamics, semiclassical approximation, Hamilton–Jacobi equation, WKB, Schrödinger equation, probability density amplitude, Bohmian mechanics
Full content is available at the original source:
arxiv.org/abs/2605.02621You Might Also Find Interesting
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